We study the effects of quenched bond disorder in frustrated easy-plane antiferromagnets in two space dimensions, using a combination of analytical and numerical techniques. We consider local-moment systems which display noncollinear long-range order at zero temperature, with the antiferromagnetic triangular-lattice XY model as a prime example. We show that (i) weak bond disorder transforms the clean ground state into a state with quasi-long-range order, and (ii) strong bond disorder results in a short-range-ordered glassy ground state. Using extensive Monte Carlo simulations, we also track the fate of the quasi-long-range-ordered phase at finite temperature and the associated Kosterlitz-Thouless-like transition. Making contact with previous studies of disordered XY models, we discuss similarities and differences concerning the interplay of disorder and frustration. Our results are also of relevance to noncollinear Heisenberg magnets in an external magnetic field.